Multidimensional Directional-Change Intrinsic Time

Stochastic volatility is an omnipresent property of all financial time series. Due to this phenomenon, equidistant time-stamps - synchronized with the flow of physical time and used to make snapshots of the market states - are either (1) too sparse and do not capture all the available high-frequency information or are (2) too dense which results in superfluous noisy events in the final time series. Returns, computed with equal time intervals, might erroneously contribute to the volatility estimation. Nevertheless, the majority of risk management tools persistently refer to physical time as the ultimate measure of financial activity (including the famous Black and Scholes equation). Directional-change intrinsic time [Guillaume et al., 1997] is one of the first endeavors to overcome the stiffness of the traditional approach and provides a new concept devoid of the mentioned shortcomings. Intrinsic time ticks when the price experiences alternating reversals of a fixed threshold Delta from local extremes. This approach is sensitive to the markets' activity and registers more ticks after the significant financial news (periods of soaring volatility) while hardly ticking over weekends. Directional-change intrinsic time was successfully applied to the liquidity estimation problem [Golub et al., 2014] and was instrumental in exploiting a Forex market trading strategy characterized by a Sharp ratio above 3 [Golub et al., 2017]. To the best of our knowledge, in all existing articles on the topic of intrinsic time, only individual time series were analyzed. The aim of our work is to extend the definition of directional-change intrinsic time to a multidimensional space, where the dimensions are formed by orthogonally placed exchange rates. The new method is tested on empirical time series from the Forex market. All details required for deriving the new algorithm are presented. Moreover, the exact description of the multidimensional dissection procedure is described in the first part of the article. We uncover two scaling laws (directional change count and overshoot move [Glattfelder et al., 2011]) in the multidimensional space and explain their dependence on the number of time series forming the space.